In any given months a man earn `A%` commission on first sale of `Rs. 1000` and further sale he earn `B%` commission. If in two following months is sales are `Rs3000` and `Rs.4000` and he earns a commission of `Rs.900` and `Rs. 1300` respectively. Find the ratio of A and B ?

Options

A. `1:3`
B. `1:5`
C. `1:4`
D. `1:8`

Answer

To solve the problem step by step, we will break down the earnings of the man based on the sales and the commissions he receives.

### Step 1: Understand the Commission Structure
The man earns:
- A% commission on the first sale of Rs. 1000.
- B% commission on any sales beyond Rs. 1000.

### Step 2: Analyze Sales and Commissions
In the first month, the total sales are Rs. 3000, and the commission earned is Rs. 900.
In the second month, the total sales are Rs. 4000, and the commission earned is Rs. 1300.

### Step 3: Calculate Commission for Month 1
For the first month (sales of Rs. 3000):
- The first Rs. 1000 earns A% commission.
- The remaining Rs. 2000 earns B% commission.

Let’s denote the commission from the first Rs. 1000 as:
\[ \text{Commission from first sale} = \frac{A}{100} \times 1000 = 10A \]

The commission from the remaining Rs. 2000 is:
\[ \text{Commission from remaining sale} = \frac{B}{100} \times 2000 = 20B \]

The total commission for the first month can be expressed as:
\[ 10A + 20B = 900 \quad \text{(1)} \]

### Step 4: Calculate Commission for Month 2
For the second month (sales of Rs. 4000):
- The first Rs. 1000 earns A% commission.
- The remaining Rs. 3000 earns B% commission.

The commission from the first Rs. 1000 remains:
\[ \text{Commission from first sale} = 10A \]

The commission from the remaining Rs. 3000 is:
\[ \text{Commission from remaining sale} = \frac{B}{100} \times 3000 = 30B \]

The total commission for the second month can be expressed as:
\[ 10A + 30B = 1300 \quad \text{(2)} \]

### Step 5: Solve the Equations
Now we have a system of equations:
1. \( 10A + 20B = 900 \)
2. \( 10A + 30B = 1300 \)

We can subtract equation (1) from equation (2):
\[ (10A + 30B) - (10A + 20B) = 1300 - 900 \]
This simplifies to:
\[ 10B = 400 \]
Thus, we find:
\[ B = 40 \]

### Step 6: Substitute B back to find A
Now, substitute \( B = 40 \) back into equation (1):
\[ 10A + 20(40) = 900 \]
This simplifies to:
\[ 10A + 800 = 900 \]
Subtracting 800 from both sides gives:
\[ 10A = 100 \]
Thus, we find:
\[ A = 10 \]

### Step 7: Find the Ratio of A and B
Now we can find the ratio of A to B:
\[ \text{Ratio of A to B} = \frac{A}{B} = \frac{10}{40} = \frac{1}{4} \]

### Final Answer
The ratio of A to B is \( \frac{1}{4} \).

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